Bayesian constraints on observable properties of exoplanetary systems using planet population synthesis simulations
Ilustraciones y tablas
- Autores:
-
Bautista Sánchez, Frank Jair
- Tipo de recurso:
- Fecha de publicación:
- 2021
- Institución:
- Universidad Nacional de Colombia
- Repositorio:
- Universidad Nacional de Colombia
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.unal.edu.co:unal/80194
- Palabra clave:
- 520 - Astronomía y ciencias afines
Planets
Planetas
Astronomy
Astronomía
Cosmic physics
Física cósmica
Bayesian inference
Planet population synthesis
Gaussian mixture model
Kernel density estimation
Formación planetaria
Mezcla gaussiana
Síntesis planetaria
Inferencia bayesiana
- Rights
- openAccess
- License
- Atribución-NoComercial 4.0 Internacional
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| dc.title.eng.fl_str_mv |
Bayesian constraints on observable properties of exoplanetary systems using planet population synthesis simulations |
| dc.title.translated.spa.fl_str_mv |
Restricciones bayesianas sobre las propiedades observables de los sistemas exoplanetarios utilizando simulaciones de síntesis de población planetaria |
| title |
Bayesian constraints on observable properties of exoplanetary systems using planet population synthesis simulations |
| spellingShingle |
Bayesian constraints on observable properties of exoplanetary systems using planet population synthesis simulations 520 - Astronomía y ciencias afines Planets Planetas Astronomy Astronomía Cosmic physics Física cósmica Bayesian inference Planet population synthesis Gaussian mixture model Kernel density estimation Formación planetaria Mezcla gaussiana Síntesis planetaria Inferencia bayesiana |
| title_short |
Bayesian constraints on observable properties of exoplanetary systems using planet population synthesis simulations |
| title_full |
Bayesian constraints on observable properties of exoplanetary systems using planet population synthesis simulations |
| title_fullStr |
Bayesian constraints on observable properties of exoplanetary systems using planet population synthesis simulations |
| title_full_unstemmed |
Bayesian constraints on observable properties of exoplanetary systems using planet population synthesis simulations |
| title_sort |
Bayesian constraints on observable properties of exoplanetary systems using planet population synthesis simulations |
| dc.creator.fl_str_mv |
Bautista Sánchez, Frank Jair |
| dc.contributor.advisor.none.fl_str_mv |
Chaparro Molano, Germán Vargas Domínguez, Santiago |
| dc.contributor.author.none.fl_str_mv |
Bautista Sánchez, Frank Jair |
| dc.subject.ddc.spa.fl_str_mv |
520 - Astronomía y ciencias afines |
| topic |
520 - Astronomía y ciencias afines Planets Planetas Astronomy Astronomía Cosmic physics Física cósmica Bayesian inference Planet population synthesis Gaussian mixture model Kernel density estimation Formación planetaria Mezcla gaussiana Síntesis planetaria Inferencia bayesiana |
| dc.subject.lemb.none.fl_str_mv |
Planets Planetas Astronomy Astronomía Cosmic physics Física cósmica |
| dc.subject.proposal.eng.fl_str_mv |
Bayesian inference Planet population synthesis Gaussian mixture model Kernel density estimation |
| dc.subject.proposal.spa.fl_str_mv |
Formación planetaria Mezcla gaussiana Síntesis planetaria Inferencia bayesiana |
| description |
Ilustraciones y tablas |
| publishDate |
2021 |
| dc.date.accessioned.none.fl_str_mv |
2021-09-15T12:53:39Z |
| dc.date.available.none.fl_str_mv |
2021-09-15T12:53:39Z |
| dc.date.issued.none.fl_str_mv |
2021-08-18 |
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Trabajo de grado - Maestría |
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info:eu-repo/semantics/masterThesis |
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info:eu-repo/semantics/acceptedVersion |
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Text |
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http://purl.org/redcol/resource_type/TM |
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acceptedVersion |
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https://repositorio.unal.edu.co/handle/unal/80194 |
| dc.identifier.instname.spa.fl_str_mv |
Universidad Nacional de Colombia |
| dc.identifier.reponame.spa.fl_str_mv |
Repositorio Institucional Universidad Nacional de Colombia |
| dc.identifier.repourl.spa.fl_str_mv |
https://repositorio.unal.edu.co/ |
| url |
https://repositorio.unal.edu.co/handle/unal/80194 https://repositorio.unal.edu.co/ |
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Universidad Nacional de Colombia Repositorio Institucional Universidad Nacional de Colombia |
| dc.language.iso.spa.fl_str_mv |
eng |
| language |
eng |
| dc.relation.references.spa.fl_str_mv |
Akaike, H. (1974). A new look at the statistical model identification. IEEE Transactions on Automatic Control, 19(6):716–723. Alibert, Y., Mordasini, C., and Benz, W. (2010). Extrasolar planet population synthesis. A&A, 526:A63. Armitage, P. J. (2009). Astrophysics of Planet Formation. Cambridge University Press. Bashi, D., Helled, R., and Zucker, S. (2018). A quantitative comparison of exoplanet catalogs. Geosciences, 8:325. Benz, W., Ida, S., Alibert, Y., Lin, D., and Mordasini, C. (2014). Planet population synthesis. In Protostars and Planets VI, page 691. Bishop, C. M. (2006). Pattern Recognition and Machine Learning (Information Science and Statistics). Springer-Verlag, Berlin, Heidelberg. Bonanos, A. Z., Yang, M., Sokolovsky, K. V., Gavras, P., Hatzidimitriou, D., Bellas-Velidis, I., Kakaletris, G., Lennon, D. J., Nota, A., White, R. L., Whitmore, B. C., Anastasiou, K. A.,Arevalo, M., Arviset, C., Baines, D., Budavari, T., Charmandaris, V., Chatzichristodoulou, C., Dimas, E., Duran, J., Georgantopoulos, I., Karampelas, A., Laskaris, N., Lianou, S., Livanis, A., Lubow, S., Manouras, G., Moretti, M. I., Paraskeva, E., Pouliasis, E., Rest, A., Salgado, J., Sonnentrucker, P., Spetsieri, Z. T., Taylor, P., and Tsinganos, K. (2019). The hubble catalog of variables (hcv). A&A, 630:A92. Brogan, C. L., P´erez, L. M., Hunter, T. R., Dent, W. R. F., Hales, A. S., Hills, R. E., Corder, S., Fomalont, E. B., Vlahakis, C., and et al. (2015). First results from high angular resolution alma observations toward the hl tau region. The Astrophysical Journal, 808(1). Brügger, N., Alibert, Y., Ataiee, S., and Benz, W. (2018). Metallicity effect and planet mass function in pebble-based planet formation models. Astronomy & Astrophysics, 619:A174. Chaparro Molano, G., Bautista, F., and Miguel, Y. (2018). Transitional disk archeology from exoplanet population synthesis. Proceedings of the International Astronomical Union, 14(S345):152–155. Chuong, D. B. and Batzoglou, S. (2008). What is the expectation maximization algorithm? Nature Biotechnology, 26(8):897–899. Claeskens, G. and Hjort, N. L. (2008). Model Selection and Model Averaging. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press. Cox, R. (1961). Algebra of Probable Inference. Johns Hopkins University Press. Crida, A. (2009). Minimum mass solar nebulae and planetary migration. The Astrophysical Journal, 698(1):606–614. Deleuil, M. and Fridlund, M. (2018). CoRoT: The First Space-Based Transit Survey to Explore the Close-in Planet Population, pages 1135–1158. Springer International Publishing. Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the em algorithm. Journal of the Royal Statistical Society. Series B (Methodological), (1):1–38. European Southern Observatory (2020). High accuracy radial velocity planet searcher. http://www.eso.org/sci/facilities/lasilla/instruments/harps.html. 2020-07-24. European Space Agency (2020). Atmospheric remote-sensing infrared exoplanet large. https://arielmission.space/. 2020-07-25. Exoplanet Exploration Program, NASA (2010). Exoplanet exploration program, NASA’s search for habitable planets and life beyond our solar system. /https://exoplanets.nasa.gov/exep/. 2020-07-21. García, E. (2019). Nonparametric statistics. https://bookdown.org/egarpor/NP-UC3M/. 2020-12-09. Gelman, A., Carlin, J., Stern, H., Dunson, D., Vehtari, A., and Rubin, D. (2013). Bayesian Data Analysis, Third Edition. Chapman & Hall/CRC Texts in Statistical Science. Taylor & Francis. Gosling, J. T. (2007). Encyclopedia of the solar system. pages 99 – 116. Academic Press, San Diego, second edition. Graziani, F., Desjarlais, M., Redmer, R., and Trickey, S. (2014). Frontiers and Challenges in Warm Dense Matter. Lecture Notes in Computational Science and Engineering. Springer International Publishing. Guidoum, A. C. (2015). Kernel estimator and bandwidth selection for density and its derivatives. Technical report. Guillot, T. and Gautier, D. (2007). Treatise on geophysics. pages 439 – 464. Elsevier, Amsterdam. Hastie, T., Tibshirani, R., and Friedman, J. (2009). The elements of statistical learning: Data mining, inference, and prediction. Springer Series in Statistics. Springer, 2nd ed. 2009. corr. 3rd printing 5th printing. edition. Hayashi, C. (1981). Structure of the solar nebula, growth and decay of magnetic fields and effects of magnetic and turbulent viscosities on the nebula. Progress of Theoretical Physics Supplement, 70:35–53. Ida, S. and Lin, D. N. C. (2004a). Toward a deterministic model of planetary formation. I. a desert in the mass and semimajor axis distributions of extrasolar planets. The Astrophysical Journal, 604(1):388–413. Ida, S. and Lin, D. N. C. (2004b). Toward a deterministic model of planetary formation. II. the formation and retention of gas giant planets around stars with a range of metallicities. The Astrophysical Journal, 616(1):567–572. Ida, S. and Lin, D. N. C. (2005). Toward a deterministic model of planetary formation. III. mass distribution of short-period planets around stars of various masses. The Astrophysical Journal, 626(2):1045–1060. Ida, S. and Lin, D. N. C. (2007). Toward a deterministic model of planetary formation IV: Effects of type-i migration. Ida, S. and Lin, D. N. C. (2008). Toward a deterministic model of planetary formation V. accumulation near the ice line. The Astrophysical Journal, 673(1):487–501. Ida, S. and Lin, D. N. C. (2010). Toward a deterministic model of planetary formation VI: Dynamical interaction and coagulation of multiple rocky embryos and super-earth systems around solar type stars. The Astrophysical Journal, 719(1):810–830. Kipping, D. M. and Sandford, E. (2016). Observational biases for transiting planets. Monthly Notices of the Royal Astronomical Society, 463(2):1323–1331. Kokubo, E. and Ida, S. (1998). Oligarchic growth of protoplanets. Icarus, 131(1):171 – 178. Lodders, K. (2003). Solar System Abundances and Condensation Temperatures of the Elements. Astrophysical Journal, 591(2):1220–1247. Makov, U. E. (2001). Mixture models in statistics. In International Encyclopedia of the Social and Behavioral Sciences, pages 9910 – 9915. Pergamon, Oxford. Mason, J. W. (2008). Exoplanets: Detection, Formation, Properties, Habitability. Springer Praxis Books / Astronomy and Planetary Sciences. Springer, 1 edition. Mayor, M., Marmier, M., Lovis, C., Udry, S., Segransan, S., Pepe, F., Benz, W., Bertaux, J. L., Bouchy, F., Dumusque, X., Lo Curto, G., Mordasini, C., Queloz, D., and Santos, N. C. (2011). The HARPS search for southern extra-solar planets occurrence, mass distribution and orbital properties of super-Earths and Neptune-mass planets. McElreath, R. and Safari, a. O. M. C. (2018). Statistical Rethinking. OCLC: 1107423386. McKinney, W. (2010). Data Structures for Statistical Computing in Python. In Stéfan van der Walt and Jarrod Millman, editors, Proceedings of the 9th Python in Science Conference, pages 56 – 61. McLachlan, G. and Peel, D. (2000). Finite Mixture Models. Wiley Series in Probability and Statistics. Wiley-Interscience. McNicholas, P. (2016). Mixture Model-Based Classification. CRC Press. Miguel, Y., Guilera, O. M., and Brunini, A. (2011a). The diversity of planetary system architectures: contrasting theory with observations. Monthly Notices of the Royal Astronomical Society, 417(1):314–332. Miguel, Y., Guilera, O. M., and Brunini, A. (2011b). The role of the initial surface density profiles of the disc on giant planet formation: comparing with observations. Monthly Notices of the Royal Astronomical Society, 412(4):2113–2124. Mordasini, C., Alibert, Y., and Benz, W. (2009a). Extrasolar planet population synthesis - I. method, formation tracks, and mass-distance distribution. A&A, 501(3):1139–1160. Mordasini, C., Alibert, Y., and Benz, W. (2009b). Extrasolar planet population synthesis ii: Statistical comparison with observation. A&A, 501(3):1161–1184. Naderi, M., Hung, W.-L., Lin, T.-I., and Jamalizadeh, A. (2019). A novel mixture model using the multivariate normal mean–variance mixture of birnbaum–saunders distributions and its application to extrasolar planets. Journal of Multivariate Analysis, 171:126 – 138. Ollivier, M. (2009). Planetary systems : detection, formation and habitability of extrasolar planets. Springer, Berlin. Parviainen, H. (2018). Bayesian methods for exoplanet science. Handbook of Exoplanets, page 1567–1590. Isella, A., Carpenter, J. M., and Sargent, A. I. (2009). Structure and evolution of pre-main-sequence circumstellar disks. The Astrophysical Journal, 701(1):260–282. Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., Vanderplas, J., Passos, A., Cournapeau, D., Brucher, M., Perrot, M., and Duchesnay, E. (2011). Scikit-learn: Machine learning in python. Journal of Machine Learning Research, 12:2825–2830. Pinilla, P., Birnstiel, T., Ricci, L., Dullemond, C. P., Uribe, A. L., Testi, L., and Natta, A. (2012). Trapping dust particles in the outer regions of protoplanetary disks. A&A, 538:A114. Pringle, J. E. (1981). Accretion discs in astrophysics. Annual Review of Astronomy and Astrophysics, 19:137–162. Ronald E. Walpole, Raymond H. Myers, S. L. M. K. E. Y. (2011). Probability and Statistics for Engineers and Scientists (9th Edition). Prentice Hall, 9 edition. Schneider, J., Dedieu, C., Le Sidaner, P., Savalle, R., and Zolotukhin, I. (2011). Defining and cataloging exoplanets: the exoplanet.eu database. A&A, 532:A79. Schwarz, G. (1978). Estimating the dimension of a model. Ann. Statist., 6(2):461–464. Scott, D. W. (1979). On optimal and data-based histograms. Biometrika, 66(3):605–610. Shalizi, C. R. (2020). Advanced Data Analysis from an Elementary Point of View. Cambridge University Press. Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis. Chapman & Hall. Sivia, D. and Skilling, J. (2006). Data Analysis: A Bayesian Tutorial. Oxford science publications. OUP Oxford. The Extrasolar Planets Encyclopaedia (1995). The extrasolar planets encyclopaedia kernel description. http://exoplanet.eu/. 2020-05-16. Toomre, A. (1964). On the gravitational stability of a disk of stars. APJ, 139:1217–1238. Udry, S. (2010). Detection and characterization of exoplanets: from gaseous giants to superearths. In In the Spirit of Lyot 2010, page E11. VanderPlas, J. (2013). Kernel density estimation in python. https://jakevdp.github.io/blog/2013/12/01/kernel-density-estimation/. 2020-12-09. VanderPlas, J. (2016). 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Advanced fiber-optic echelle (OFOE). urlhttps://www.eso.org/public/teles-instr/alma/. 2020-07-23. Williams, J. P. and Cieza, L. A. (2011). Protoplanetary disks and their evolution. Annual Review of Astronomy and Astrophysics, 49(1):67–117. Wright, J. and Gaudi, B. (2012). Exoplanet detection methods. Planets, Stars and Stellar Systems. Volume 3: Solar and Stellar Planetary Systems. Zakamska, N., Pan, M., and Ford, E. (2010). Observational biases in determining extrasolar planet eccentricities in single-planet systems. Monthly Notices of the Royal Astronomical Society, 410. Zucchini, W., Berzel, A., and Nenadic, O. (2003). Applied smoothing techniques. Part I: Kernel Density Estimation, 15. |
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Universidad Nacional de Colombia |
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Bogotá - Ciencias - Maestría en Ciencias - Física |
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Departamento de Física |
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Facultad de Ciencias |
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Bogotá, Colombia |
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Atribución-NoComercial 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Chaparro Molano, Germán08c4d569704ab20c98aa0e3806591182Vargas Domínguez, Santiagoe9d25dbef9da29422adab51ba9732510600Bautista Sánchez, Frank Jair16ca30ee5510a2249dd9adaa8366b8642021-09-15T12:53:39Z2021-09-15T12:53:39Z2021-08-18https://repositorio.unal.edu.co/handle/unal/80194Universidad Nacional de ColombiaRepositorio Institucional Universidad Nacional de Colombiahttps://repositorio.unal.edu.co/Ilustraciones y tablasRecent advances in exoplanet observations have led to the discovery of nearly 4400 planets in more than 3000 planetary systems. However, many physical properties of these systems remain largely unknown due to observational biases and stochasticity in their formation processes. We address current limitations of exoplanet classification by formulating a new, robust classification scheme based on the first moment of planet mass vs. semi-major axis distribution using Gaussian Mixture Models (GMM). We validate our method via Approximate Bayesian Computation and Information Criteria, which shows that it is robust with uncertainties measurement and adaptable to new observations. Our scheme yields four planetary system classes: Sub-Mercurial systems, Venusian systems, Solar-like systems, and Periphery systems. We also propose a new method for compensating observational biases while addressing stochasticity in star and planet formation. To this end, we develop a Bayesian probabilistic formalism in which we take priors from observed planetary systems and marginalize them over synthetic likelihood functions. It allows us to estimate probability distribution functions for variables of interest, such as the total number of planets, total planetary mass, rocky planetary mass, center of mass, among others. We generate our synthetic likelihood functions from a multivariate Kernel Density Estimation (KDE) model based on the results of a Monte Carlo simulation of 1200 planet population synthesis models, drawn from observational priors obtained from the literature. We assess the performance of the kernel parameter choice using cross-validation. Therefore, we got probability distributions of physical variables of interest for ten observed systems using data from public catalogues. For the selected systems, we infer that they had initial disks with masses around 0.1 M⊙ ± 0.01 M⊙, their centers of mass are located around 5 AU ± 2 AU, and they should have around seven more planets than are currently observed. We also conclude that the number of rocky planets significantly contributes to the total number of planets, so we expect to find more rocky planets in future observations. Our formalism allows getting the probability distributions of exoplanetary systems unobserved or with biased properties. It will help steering future astronomical surveys and motivating further questions of observed planetary systems.Los avances recientes en las observaciones de exoplanetas han llevado al descubrimiento de casi 4400 planetas en más de 3000 sistemas planetarios. Sin embargo, muchas propiedades físicas de estos sistemas siguen siendo en gran parte desconocidas debido a sesgos observacionales y estocasticidad en sus procesos de formación. Abordamos las limitaciones actuales de la clasificación de exoplanetas mediante la formulación de un nuevo y robusto esquema de clasificación basado en el primer momento de la masa del planeta frente a la distribución del eje semi-mayor utilizando modelos de mezcla gaussianos (GMM). Validamos nuestro método mediante criterios de información y cálculo bayesianos aproximados, lo que demuestra que es robusto con la medición de incertidumbres y adaptable a nuevas observaciones. Nuestro esquema produce cuatro clases de sistemas planetarios: sistemas Sub-Mercurianos, sistemas Venusianos, sistemas similares al solar y sistemas periféricos. También proponemos un nuevo método para compensar los sesgos de observación al abordar la estocasticidad en la formación de estrellas y planetas. Con este fin, desarrollamos un formalismo probabilístico bayesiano en el que tomamos información previa de los sistemas planetarios observados y los marginalizamos sobre las funciones de verosimilitud sintéticas. Lo que nos permite estimar funciones de distribución de probabilidad para variables de interés, como el número total de planetas, la masa total planetaria, la masa planetaria rocosa, centro de masa, entre otras. Generamos nuestras funciones de verosimilitud sintéticas a partir de un modelo multivariado de estimación de densidad de núcleo (KDE) basado en los resultados de una simulación de Monte Carlo de 1200 modelos de síntesis de población planetaria, extraídos de observaciones previas obtenidas de la literatura. Evaluamos el rendimiento de la elección del parámetro de una función núcleo (Kernel) mediante validación cruzada. De esta manera, obtuvimos distribuciones de probabilidad de variables físicas de interés para diez sistemas observados utilizando datos de catálogos públicos. Para los sistemas seleccionados, inferimos que tenían discos iniciales con masas alrededor 0.1 M⊙ ± 0.01 M⊙, sus centros de masa se ubican alrededor de 5 AU ± 2 AU, y deberían tener alrededor de siete planetas más de los que se observan actualmente. También concluimos que el número de planetas rocosos contribuye significativamente al número total de planetas, por lo que esperamos encontrar más planetas rocosos en futuras observaciones. Nuestro formalismo permite obtener las distribuciones de probabilidad de sistemas exoplanetarios no observados o con propiedades sesgadas. Este formalismo ayudará a dirigir los estudios astronómicos futuros y a motivar más preguntas sobre los sistemas planetarios observados. (Texto tomado de la fuente).Incluye anexosMaestríaMagíster en Ciencias - FísicaFormación Eexoplanetaria - Sintesis planetariaxii, 92 páginasapplication/pdfengUniversidad Nacional de ColombiaBogotá - Ciencias - Maestría en Ciencias - FísicaDepartamento de FísicaFacultad de CienciasBogotá, ColombiaUniversidad Nacional de Colombia - Sede Bogotá520 - Astronomía y ciencias afinesPlanetsPlanetasAstronomyAstronomíaCosmic physicsFísica cósmicaBayesian inferencePlanet population synthesisGaussian mixture modelKernel density estimationFormación planetariaMezcla gaussianaSíntesis planetariaInferencia bayesianaBayesian constraints on observable properties of exoplanetary systems using planet population synthesis simulationsRestricciones bayesianas sobre las propiedades observables de los sistemas exoplanetarios utilizando simulaciones de síntesis de población planetariaTrabajo de grado - Maestríainfo:eu-repo/semantics/masterThesisinfo:eu-repo/semantics/acceptedVersionTexthttp://purl.org/redcol/resource_type/TMAkaike, H. (1974). A new look at the statistical model identification. IEEE Transactions on Automatic Control, 19(6):716–723.Alibert, Y., Mordasini, C., and Benz, W. (2010). Extrasolar planet population synthesis. A&A, 526:A63.Armitage, P. J. (2009). Astrophysics of Planet Formation. Cambridge University Press.Bashi, D., Helled, R., and Zucker, S. (2018). A quantitative comparison of exoplanet catalogs. Geosciences, 8:325.Benz, W., Ida, S., Alibert, Y., Lin, D., and Mordasini, C. (2014). Planet population synthesis. In Protostars and Planets VI, page 691.Bishop, C. M. (2006). Pattern Recognition and Machine Learning (Information Science and Statistics). Springer-Verlag, Berlin, Heidelberg.Bonanos, A. Z., Yang, M., Sokolovsky, K. V., Gavras, P., Hatzidimitriou, D., Bellas-Velidis, I., Kakaletris, G., Lennon, D. J., Nota, A., White, R. L., Whitmore, B. C., Anastasiou, K. 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Part I: Kernel Density Estimation, 15.EstudiantesInvestigadoresMaestrosPúblico generalORIGINAL1022973370.2021.pdf1022973370.2021.pdfTesis de Maestría en Ciencias - Físicaapplication/pdf6604536https://repositorio.unal.edu.co/bitstream/unal/80194/5/1022973370.2021.pdf113caf7e517eb04d6076bf3c40cfb035MD55LICENSElicense.txtlicense.txttext/plain; charset=utf-83964https://repositorio.unal.edu.co/bitstream/unal/80194/3/license.txtcccfe52f796b7c63423298c2d3365fc6MD53THUMBNAIL1022973370.2021.pdf.jpg1022973370.2021.pdf.jpgGenerated Thumbnailimage/jpeg3931https://repositorio.unal.edu.co/bitstream/unal/80194/6/1022973370.2021.pdf.jpg73e29e88d66e8e056d18ff8992b9a118MD56unal/80194oai:repositorio.unal.edu.co:unal/801942024-07-29 00:02:57.808Repositorio Institucional Universidad Nacional de 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